Problem

Source: Iranian National Olympiad (3rd Round) 2003

Tags: algebra, polynomial, induction, algebra proposed



Let $ c\in\mathbb C$ and $ A_c = \{p\in \mathbb C[z]|p(z^2 + c) = p(z)^2 + c\}$. a) Prove that for each $ c\in C$, $ A_c$ is infinite. b) Prove that if $ p\in A_1$, and $ p(z_0) = 0$, then $ |z_0| < 1.7$. c) Prove that each element of $ A_c$ is odd or even. Let $ f_c = z^2 + c\in \mathbb C[z]$. We see easily that $ B_c: = \{z,f_c(z),f_c(f_c(z)),\dots\}$ is a subset of $ A_c$. Prove that in the following cases $ A_c = B_c$. d) $ |c| > 2$. e) $ c\in \mathbb Q\backslash\mathbb Z$. f) $ c$ is a non-algebraic number g) $ c$ is a real number and $ c\not\in [ - 2,\frac14]$.